Notes — Chapter 1: An Introduction to Geometry

To the Student. Your study of geometry builds on intuitive understandings: for example, of how rulers are used and of how braces can steady a framework. Geometry gives you tools for systematically solving problems such as those of the Surfer and Spotter. This chapter shows how problem-solving led to the development of geometry and how geometry, in turn, helps solve problems. Here we first meet the geometric terms and ideas that are used throughout the text.

The final lesson, We Can’t Go on Like This, shows the need for a more systematic approach. In geometry examples let you see what is going on — what works and what doesn’t. But you need tools to help you discover things, to avoid fooling yourself, and to back up your insights. These you will master as you work through this book. Revisit the Surfer, the Spotter, and other problems. You will find simpler approaches and new understandings as you move ahead. _

Lesson 1. Lines in Designing a City. The terms point, line, line segment, and plane are informally introduced, followed by definitions of several terms relating these basic geometric concepts: collinear and noncollinear, coplanar, and concurrent.

Interesting configurations of lines are included in the exercises. Students are asked to look at figures, identify and count their components, and draw other figures that lead to results that are both puzzling and surprising. These teach the basics and will suggest a world of possibilities to the student whose curiosity is aroused.

Lesson 2. Angles in Measuring the Earth. The terms ray and angle are informally introduced and the use of a protractor to measure angles is reviewed.

The exercises ask the students to make visual judgments followed by practice in making accurate measurements to verify them.

Lesson 3. Polygons and Polyhedra: Pyramid Architecture. The terms polygon and polyhedron are informally introduced, together with the names of the most commonly encountered polygons. The concepts of perimeter and area are reviewed.

The exercises range from questions about pentominoes to building a model of the Great Pyramid to a remarkable dissection problem (for the latter see also under Set III of Lesson 1, Chapter 7).

Lesson 4. Constructions: Telling Time with Shadows. The idea of a construction is introduced and the methods for carrying out two basic constructions, bisecting a line segment and bisecting an angle, are introduced without proof.

In the exercises, constructions are applied to finding north with shadows, determining the hour spokes of

a simple sundial, and discovering something surprising about bisecting the angles of a rectangle.

Lesson 5. We Can’t Go on Like This. The concluding lesson of this chapter emphasizes that our thus-far informal approach to geometry has led to interesting as well as practical results but is ultimately unsatisfactory because we have no convincing basis for knowing that the conclusions we have been drawing are necessarily correct.

The exercises show the need for critical thinking and careful analysis. They are also provocative: Why is it easy to trisect a line segment, yet impossible to trisect an angle with only straightedge and compass? Why do several consecutive figures showing regions formed in circles seem to establish a simple pattern, after which the pattern fails?